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What is the fundamental basis of resolution of vector. Suppose we have a vector $\vec{mg}$, now we resolve it into two components, horizontal and vertical. My question is what is the basis for telling that this is the horizontal and this is the vertical component since the Cartesian plane can be drawn on any direction.

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Ok.. another question why was an electron's charge called negative and proton's positive? – Sreekanth Karumanaghat Apr 10 '13 at 12:47

You're right that the co-ordinates can be written in any direction. But, there's always a left-right (horizontal) co-ordinate, bottom-top (vertical) co-ordinate and out-in (normal) co-ordinate. You can choose any of these co-ordinates as $x$, $y$ and $z$. But at any given time, there's gonna be an axis for each of these co-ordinates. It's not a rule that $x$ axis should be horizontal and $y$ should be vertical
(which was followed in my high-school, and is declared to be true).

If you know how to use your math skillfully, it's very easy. In this case when we consider classical mechanics, weight is always taken downwards. So, it's in the vertical (negative) direction. There's no horizontal component. Even if you try to resolve, it yields zero..!

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The cartesian co-ordiante can be drawn in any direction. But the two co-ordinate axes must be perpendicular together. Consider the force is resolved in x axis and and the value is equal to the product of magnitude of force and cosine of the angle made by the force and x axis if you consider the weight it will always acts in downward direction. And it dont have an horizontal component because the angle between the weight acting downward and horizontal direction is 90 and it cosine turn out to be zero there is no horizontal component

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